A heavy group is one that is too important to sacrifice, but which requires so much defence that defending it will result in a significant loss.
See also older discussion, now at heavy versus overconcentrated.
Heavy play usually counts as a mistake. Perhaps the most convincing general reason to play heavily is to save cutting stones, as part of a divide and conquer strategy; in that case the point is that your heaviness is balanced by compensation in the form of your opponent's weak group(s).
See Heavy Examples.
See also /Discussion.